User:Kensteve/Sandbox

$$\frac{\Gamma \vdash \alpha \rightarrow \beta \qquad \Gamma \vdash \alpha}{\Gamma \vdash \beta}\textit{Modus~Ponens}$$

$$\displaystyle\frac{}{\displaystyle\frac{A}{\displaystyle\frac{B}{C}}}$$

$$\frac{}{\frac{A}{\frac{B}{C}}}$$

$$\displaystyle\frac{}{\displaystyle\frac{\Gamma x:~\alpha \rightarrow \beta \rightarrow \gamma~ \vdash x:~\alpha \rightarrow \beta \rightarrow \gamma \qquad \Gamma z:~\alpha~ \vdash z:~\alpha}{\displaystyle\frac{\Gamma x:~\alpha \rightarrow \beta \rightarrow \gamma z:~\alpha~ \vdash xz:\beta \rightarrow \gamma}{C}}}$$

$$\displaystyle\frac{} \displaystyle\frac{\displaystyle\frac{\Gamma x:~\alpha \rightarrow \beta \rightarrow \gamma~ \vdash x:~\alpha \rightarrow \beta \rightarrow \gamma \qquad \Gamma z:~\alpha~ \vdash z:~\alpha}{\Gamma x:~\alpha \rightarrow \beta \rightarrow \gamma, z:~\alpha~ \vdash xz:\beta \rightarrow \gamma}(\rightarrow E) \qquad \displaystyle\frac{\Gamma y:~\beta \rightarrow \gamma~ \vdash y:~\beta \rightarrow \gamma \qquad \Gamma z:~\alpha~ \vdash z:~\alpha}{\Gamma y:~\alpha \rightarrow \beta, z:~\alpha~ \vdash yz:\beta}(\rightarrow E)} {\displaystyle\frac{{\displaystyle\frac{\Gamma x:~\alpha \rightarrow \beta \rightarrow \gamma,~ y:~\alpha \rightarrow \beta,~ z:~\alpha \vdash xz(yz):~ \alpha}{\displaystyle\frac{\Gamma x:~\alpha \rightarrow \beta \rightarrow \gamma,~ y:~\alpha \rightarrow \beta \vdash \lambda z.xz(yz):~ \alpha}{\Gamma, x:~\alpha \rightarrow \beta \rightarrow \gamma \vdash \lambda yz.xz(yz):~(\alpha \rightarrow \beta) \rightarrow \alpha}(\rightarrow I)}}(\rightarrow I)}{\vdash \lambda xyz.xz(yz):~(\alpha \rightarrow \beta \rightarrow \gamma) \rightarrow (\alpha \rightarrow \beta) \rightarrow \alpha}(\rightarrow I)}$$

$$\displaystyle\frac{}{\displaystyle\frac{x:\alpha \vdash x:\alpha}{\vdash \lambda x.x:\alpha \to \alpha}(2)}(1)$$

$$\displaystyle\frac{}{\displaystyle\frac{\displaystyle\frac{x:\alpha \vdash x:\alpha}{\vdash \lambda x.x:\alpha \to \alpha}}}{A}$$